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Qualitative Theory of Dynamical Systems

, Volume 1, Issue 2, pp 211–230 | Cite as

On the vanishing set of inverse integrating factors

  • Lucio R. Berrone
  • Hector J. Giacomini
Article

Abstract

We study in this paperC 1 two-dimensional dynamical systems of the formx=P(x,y), y=Q(x,y). We analyse the properties of the vanishing set of inverse integrating factors V, which are defined asC 1 solutions of the equation\(P\frac{{\partial V}}{{\partial x}} + Q\frac{{\partial V}}{{\partial y}} = Vdiv(P,Q)\). Isolated zeros of V are studied and their relationships with critical points of the system is evidenced. We show how the knowledge of an inverse integrating factor in a neighborhood of a critical point provides useful information on the local dynamics of the system. A general result is proved on vanishing of V on the separatrix curves of a saddle-point. Finally, the problem of vanishing on graphs of inverse integrating factors is discussed. It is shown that a bounded graph is contained in the vanishing set of an inverse integrating factor when the critical points of the graph are non-degenerate.

Keywords

Local Dynamic Closed Orbit Invariant Solution Saddle Connection Separatrix Curve 
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Copyright information

© Birkhäuser-Verlag 2000

Authors and Affiliations

  1. 1.Departamento de MatemáticaCONICETRosarioArgentina
  2. 2.Laboratoire de Mathématique et Physique Théorique, CNRS (UPRES-A 6083), Faculté des Sciences et TechniquesUniversité de ToursToursFrance

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